Many problems in the design and implementation of computational schemes may be studied using the theory and methods of mathematical programming. One seeks to minimize bounds for the errors in the calculated results ob...
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Many problems in the design and implementation of computational schemes may be studied using the theory and methods of mathematical programming. One seeks to minimize bounds for the errors in the calculated results obtained from a given set of input data, exploiting analytical relations. We describe optimal quadrature rules and give an application to the evaluation of the sums of power series, belonging to an important class. We present results which are based on the theory of linear and semi-infinite programming. We also study the associated complexity issues and obtain simple qualitative results for the computational work required. (C) 2006 Elsevier B.V. All rights reserved.
Converse Kuhn-Tucker and duality results for constrained continuous programming, under generarlized convex hypotheses, have often been proved independently of the known results for programming in finite dimensions. It...
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Converse Kuhn-Tucker and duality results for constrained continuous programming, under generarlized convex hypotheses, have often been proved independently of the known results for programming in finite dimensions. It is shown here that a large class of such results follow immediately from known results with the standard definitions of pseudoconvex, quasiconvex, pseudoinvex, and quasiinvex, by using Frechet derivatives and integral formulas describing them. Both Wolfe and Mond-Weir duals are included, and both pointwise and integral constraints in a continuous program. Also sufficient conditions, simpler than those previously given, are obtained for a minimax problem, or a minsup problem of continuous programming.
In this work, we study continuous reformulations of zero-one concave programming problems. We introduce new concave penalty functions and we prove, using general equivalence results here derived, that the obtained con...
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In this work, we study continuous reformulations of zero-one concave programming problems. We introduce new concave penalty functions and we prove, using general equivalence results here derived, that the obtained continuous problems are equivalent to the original combinatorial problem.
The usual duality results are established for static symmetric dual fractional programming problems without nonnegativity constraints using the notion of invexity which has allowed weakening various types of convexity...
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The usual duality results are established for static symmetric dual fractional programming problems without nonnegativity constraints using the notion of invexity which has allowed weakening various types of convexity assumptions for mathematical programming problems extensively studied in the literature. These results are also extended to their continuous analogous as dynamic generalizations.
In this work, we study continuous reformulations of zero-one programming problems. We prove that, under suitable conditions, the optimal solutions of a zero-one programming problem can be obtained by solving a specifi...
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In this work, we study continuous reformulations of zero-one programming problems. We prove that, under suitable conditions, the optimal solutions of a zero-one programming problem can be obtained by solving a specific continuous problem.
The problem of optimal transportation was formalized by the French mathematician Gaspard Monge in 1781. Since Kantorovitch, this (generalized) problem is formulated with measure theory. Based on Interval Arithmetic, w...
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The problem of optimal transportation was formalized by the French mathematician Gaspard Monge in 1781. Since Kantorovitch, this (generalized) problem is formulated with measure theory. Based on Interval Arithmetic, we propose a guaranteed discretization of the Kantorovitch's mass transportation problem. Our discretization is spatial: supports of the two mass densities are partitioned into finite families. The problem is relaxed to a finite dimensional linear programming problem whose optimum is a lower bound to the optimum of the initial one. Based on Kantorovitch duality and Interval Arithmetic, a method to obtain an upper bound to the optimum is also provided. Preliminary results show that good approximations are obtained.
Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in ...
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Optimality conditions and duality results are obtained for a class of control problems having a nondifferentiable term in the integrand of the objective functional. These results generalize many well-known results in optimal control theory involving differentiable functions, and also provide a relationship with certain nondifferentiable mathematical programming problems. Some extensions concerning the unified treatment of optimal control theory and continuous programming are also mentioned. Finally, a control problem containing an arbitrary norm, along with its appropriate norm, is given.
In this work, we study exact continuous reformulations of nonlinear integer programming problems. To this aim, we preliminarily state conditions to guarantee the equivalence between pairs of general nonlinear problems...
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In this work, we study exact continuous reformulations of nonlinear integer programming problems. To this aim, we preliminarily state conditions to guarantee the equivalence between pairs of general nonlinear problems. Then, we prove that optimal solutions of a nonlinear integer programming problem can be obtained by using various exact penalty formulations of the original problem in a continuous space.
This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2)...
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This paper presents an approximation scheme for optimal control problems using finite-dimensional linear programs and interval analysis. This is done in two parts. Following Vinter approach (SIAM J Control Optim 31(2):518-538, 1993) and using occupation measures, the optimal control problem is written into a linear programming problem of infinite-dimension (weak formulation). Thanks to Interval arithmetic, we provide a relaxation of this infinite-dimensional linear programming problem by a finite dimensional linear programming problem. A proof that the optimal value of the finite dimensional linear programming problem is a lower bound to the optimal value of the control problem is given. Moreover, according to the fineness of the discretization and the size of the chosen test function family, obtained optimal values of each finite dimensional linear programming problem form a sequence of lower bounds which converges to the optimal value of the initial optimal control problem. Examples will illustrate the principle of the methodology.
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